Peak congestion in multi-server service systems with slowly varying arrival rates

William A. Massey, Ward Whitt

Research output: Contribution to journalArticlepeer-review

32 Scopus citations


In this paper we consider the Mt/G/∞ queueing model with infinitely many servers and a nonhomogeneous Poisson arrival process. Our goal is to obtain useful insights and formulas for nonstationary finite-server systems that commonly arise in practice. Here we are primarily concerned with the peak congestion. For the infinite-server model, we focus on the maximum value of the mean number of busy servers and the time lag between when this maximum occurs and the time that the maximum arrival rate occurs. We describe the asymptotic behavior of these quantities as the arrival changes more slowly, obtaining refinements of previous simple approximations. In addition to providing improved approximations, these refinements indicate when the simple approximations should perform well. We obtain an approximate time-dependent distribution for the number of customers in service in associated finite-server models by using the modified-offered-load (MOL) approximation, which is the finite-server steady-state distribution with the infinite-server mean serving as the offered load. We compare the value and lag in peak congestion predicted by the MOL approximation with exact values for Mt/M/s delay models with sinusoidal arrival-rate functions obtained by numerically solving the Chapman-Kolmogorov forward equations. The MOL approximation is remarkably accurate when the delay probability is suitably small. To treat systems with slowly varying arrival rates, we suggest focusing on the form of the arrival-rate function near its peak, in particular, on its second and third derivatives at the peak. We suggest estimating these derivatives from data by fitting a quadratic or cubic polynomial in a suitable interval about the peak.

Original languageAmerican English
Pages (from-to)157-172
Number of pages16
JournalQueueing Systems
Issue number1-4
StatePublished - Jun 1997
Externally publishedYes

ASJC Scopus subject areas

  • Statistics and Probability
  • Computer Science Applications
  • Management Science and Operations Research
  • Computational Theory and Mathematics


  • Infinite-server queues
  • Modified-offered-load approximation
  • Multi-server queues
  • Nonstationary queues
  • Peak congestion
  • Slowly varying arrival rates
  • Time lag
  • Time-dependent arrival rates
  • Uniform acceleration expansions


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